Let $(\Omega,M,\mu)$ be a $\underline{\text{signed}}$ measure space. Is it true that the definition of signed measure implicitly implies that
Either for any sequence of disjoint measurable sets with negative measure $\{E_i\}$, $\sum\limits_{i=1}^{+\infty}\mu(E_i)$ is finite, or for any disjoint sequence of measurable sets with positive measure $\{E_i\}$, $\sum\limits_{i=1}^{+\infty}\mu(E_i)$ is finite
?
Motivation:
Many textbooks describe the additivity in the definition of signed measure $\mu$ on $(\Omega, M)$ (where $M$ is a $\sigma$-algebra on $\Omega$) as:
If $E_i$ are disjoint measurable sets, then $\mu\left( \bigcup\limits_{i=1}^{+\infty}E_i\right)=\sum\limits_{i=1}^{+\infty}\mu\left(E_i\right)\cdots (\star) $, and $\mu\left( \bigcup\limits_{i=1}^{+\infty}E_i\right)$ is finite $\implies \sum\limits_{i=1}^{+\infty}\mu\left(E_i\right)$ converges absolutely.
I am confused about what may happen if $\mu\left( \bigcup\limits_{i=1}^{+\infty}E_i\right)$ is $+\infty$ or $-\infty$. I think the definition doesn't say $ (\star)$ only holds when $\mu\left( \bigcup\limits_{i=1}^{+\infty}E_i\right)$ is finite, i.e. it is also correct if $\mu\left( \bigcup\limits_{i=1}^{+\infty}E_i\right)$ is $+\infty$ or $-\infty$. But in the latter case, $\sum\limits_{i=1}^{+\infty}\mu\left(E_i\right)$ is not necessarily absolutely convergent, and the Riemman's rearrangement theorem may cause some trouble. I looked up some post on this site, and it seems the definition itself implies $\sum\limits_{i=1}^{+\infty}\mu\left(E_i\right)$ is always well-defined, i.e. the sum will not change under any arrangement, which may happen if and only if at least one of $\sum\limits_{\mu(E_i)<0}\mu\left(E_i\right)$ and $\sum\limits_{\mu(E_i)>0}\mu\left(E_i\right)$ is finite.
So can we conclude:
Let $(\Omega,M,\mu)$ be a signed measure space. Then either for any sequence of disjoint measurable sets with negative measure $\{E_i\}$, $\sum\limits_{i=1}^{+\infty}\mu(E_i)$ is finite, or for any disjoint sequence of measurable sets with positive measure $\{E_i\}$, $\sum\limits_{i=1}^{+\infty}\mu(E_i)$ is finite
?
Suppose $\mu (E)<\infty$ for all $E$. Let $(E_i)$ be a disjoint sequence of measurable sets with positive measure. Then $\sum\limits_{i=1}^{+\infty}\mu(E_i)$ is finite by the very definition if a signed measure.
Similarly, if $\mu (E)>-\infty$ for all $E$ and $(E_i)$ be a disjoint sequence of measurable sets with negative measure then $\sum\limits_{i=1}^{+\infty}(-\mu(E_i))$ is finite which measn $\sum \mu(E_i)$ converges.
A signed measure is not allowed to take on both of the values $\pm \infty$.